Answer
See explanation.
Work Step by Step
We require that $\Phi(\phi)=\Phi(\phi+2 \pi)$.
$\Phi(\phi+2 \pi)=e^{im_l(\phi+2 \pi)}= e^{im_l\phi} e^{im_l(2 \pi)}= \Phi(\phi) e^{im_l(2 \pi)}$
For the desired expression to be true, it is required that $e^{im_l(2 \pi)}=1$
Euler's formula states:
$e^{im_l(2 \pi)}=cos(m_l(2 \pi))+i\;sin(m_l(2 \pi))$
Therefore, an inspection of the cosine graph reveals that $e^{im_l(2 \pi)}=1$ if and only if $m_l$ is an integer.
